Another popular measure for tournaments, the cutwidth was introduced
by Chudnovsky, Fradkin and Seymour [1]} to study tournament immersions.
The vertex ordering used to certify that cutwidth is bounded
can be shown to exclude grids, meaning that it is a witness of bounded twin-width.
Hence bounded cutwidth implies bounded twin-width.
Another parameter, closely related to subdivisions in tournaments,
is pathwidth, studied by Fradkin and Seymour [2]}.
Bounded pathwidth of tournaments implies bounded cliquewidth,
which in turn also implies bounded twin-width, see [3]}.
Thus, we have the following chain of inclusions of complexity parameters
(if a parameter is bounded, all the ones listed after are also bounded):
cutwidth, pathwidth, cliquewidth, twin-width, and VC-dimension.
For more on the subject, we invite to read Fomin and Pilipczuk [4]}
or Pilipczuk's thesis [5]}.
